Theorem  can be understood as saying that if there is an inefficiency in a small part of the optimal coupling (represented by \(\tilde{\gamma }\) ), then the larger plan \(\gamma \)  cannot be optimal and efficient.
The proof, as given in [1]}, follows this line of reasoning.
Theorem  is used to prove the Theorem , the Kantorovich Duality Theorem. The Kantorovich Duality Theorem establishes the structure of optimality, which is that the support must be \(c\) -cyclically monotone.
Additionally, it is a model for Corollary , which depends on Theorem  via Theorem  (Kantorovich Duality Theorem).
